Climate change,hydrodynamical models & extreme sea levels

Adam ButlerJanet HeffernanJonathan Tawn Lancaster University Department of Mathematics & Statistics

The problem

Introduction

Understanding the impacts of climate change upon extreme sea levels.

Understanding spatial variation in impacts.

Use statistical ideas of spatial statistics and extreme value theory (Smith, 2002).

Attempt to build physically realistic models.

Applications: flood defence, offshore engineering, insurance…

Hydrodynamical models

POL models < 35km NEAC grid <

12km NISE grid V V

Observed climate inputs

Run using observational climate data. Model run for period 1970-1999. Run on NICE and NEAC grids.

Reasonable fit to observational data (Flather, 1987).

Test for evidence for a linear temporal trend in extreme values.

Climate sensitivity

Generate 30-year long sequences of model output under two hypothetical climate scenarios:

1 “Current” CO2 levels2 Double “current” CO2 levels

Sequences are stationary. Climate inputs generated using the ECHAM-4 climate model.

We will construct parametric models. Interest is in comparing the parameter estimates obtained

under the two scenarios.

Univariate extremes

The GEV distribution

Blockwise maxima converge to a GEV (Generalized Extreme Value) distribution:

is the shape parameter.

Conditions for convergence include:• independence or weak dependence • stationarity (Leadbetter, 1987).

/1

)(1exp)(

xxF

Modelling extremes

General ideas Ignore distribution of original data. Can model maxima using GEV distribution. Alternative approaches to extremes exist: e.g. threshold methods (Coles, 2001).

Application to the POL data Model the annual maxima at each site. Assume independence between sites.

Previous findings Surge residuals

Changes (cm) in 50 year surge levels for the NISE grid.

Estimates exhibit spatial variability.

Spatial extremes

Multivariate extremes Componentwise maxima Multivariate Extreme Value Distribution Nonparametric or parametric estimation ?

Parametric approaches Marginal and dependence characteristics. The Multivariate Logistic Distribution

Alternative parametric models (Tawn, ?) Physically motivated subsets

d

iixxF

1

/1exp)(

Multivariate logistic distribution

Multivariate logistic distribution

Multivariate logistic distribution

Multivariate logistic distribution

Spatial extremes

• Assume smooth spatial variation in GEV parameters.

• This implies spatial coherence. Assume that observations at neighbouring sites are

spatially dependent. Use a multivariate approach to extremes, with one

dimension for each site.

Benefits Improved efficiency in parameter estimation. Interpretable estimates of spatial structure. Allows extrapolation to ungauged sites. Enables regional-level estimates to be derived.

Current work

Marginal or joint estimation?

Bivariate case, GEV margins, logistic dependence.

Three possible methods for estimation: • joint likelihood (“joint”),• product of marginal likelihoods (“marginal”)• robust version of “marginal” approach

Marginal approach results in reduced efficiency if there is dependence.

How large is this effect ? Simulation study.

See: Shi, Smith & Coles (1992), Barao & Tawn (1999).

Conclusions

Further work “Bivariate efficiency” study Comparison of approaches to spatial extremes POL data: extremal trends POL data: climate sensitivity Scottish rainfall data…?

Statistical significance Application of modern extreme value methods in an

applied context High dimensionality

Questions ?